Hecke algebra on homogeneous trees and relations with Toeplitz and Hankel operators

Author:
Janusz Wysoczański

Journal:
Proc. Amer. Math. Soc. **122** (1994), 1203-1210

MSC:
Primary 46J30; Secondary 05C05, 47B35, 47D30

MathSciNet review:
1213871

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Abstract: We consider the Hecke algebra on homogeneous trees. We prove that it is a maximal abelian subalgebra of some operator algebras if the degree of the tree is greater than 2. There we show the influence of geometry of the tree on that fact. If the degree is 2 (for example, in the case of integers) then we show that operators which commute with the Hecke algebra can be uniquely represented as a sum of Hankel and Toeplitz matrices.

**[C]**P. Cartier,*Harmonic analysis on trees*, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 419–424. MR**0338272****[P]**T. Pytlik,*Radial functions on free groups and a decomposition of the regular representation into irreducible components*, J. Reine Angew. Math.**326**(1981), 124–135. MR**622348**, 10.1515/crll.1981.326.124**[S]**Jean-Pierre Serre,*Trees*, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR**607504****[R]**Jean Renault,*A groupoid approach to 𝐶*-algebras*, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR**584266**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1994-1213871-3

Keywords:
Hecke algebra,
maximal abelian subalgebra,
homogeneous tree,
Hankel and Toeplitz matrices

Article copyright:
© Copyright 1994
American Mathematical Society